**Alfredo Capelli** was an Italian mathematician who worked on group theory and independently discovered Sylov's theorem.

- Capelli had published two notes on the theory of groups in the same journal two years earlier while he was working on his thesis supervised by Battaglini.
- These were independent discoveries by Capelli since he was unaware of Sylow's 1872 paper and Netto's 1874 paper where these results were first published.
- However, Capelli's paper also contains standard results later proved by Frobenius and Burnside which have been attributed to these later authors.
- Capelli graduated from the University of Rome in 1877 and then continued to develop his mathematical skills working as Felice Casorati's assistant at the University of Pavia.
- Casorati corresponded regularly with Weierstrass and had developed a strong link between Italian and German mathematicians so Capelli's visit to Berlin was a natural one in this context.
- Over this period, Capelli did not work on group theory but his interests had moved to the theory of algebraic forms.
- In Palermo, the situation finally began to change in 1878 with the arrival of Cesare Arzelà (1847-1912), who held the chair of algebra for two years, and of Capelli, who replaced Arzelà when the latter moved to Bologna.
- In 1884 Capelli published another major work on group theory Sopra la composizione dei gruppi di sostituzioni Ⓣ(On the composition of groups of substitutions).
- Why, one might ask is it not then known today as the 'Capelli argument'.
- Capelli and Frattini had both been students of Battaglini, with Frattini graduating two years before Capelli.
- He also used the beautiful 'Frattini argument', although he clearly stated that this was due to Capelli.
- Perhaps Frattini should have used the term 'Capelli argument' himself as this would have guaranteed Capelli receiving credit.
- While at Palermo, Capelli collaborated with Giovanni Garbieri (1847-1931) who had succeeded Giusto Bellavitis at the University of Padua in 1882.
- Capelli had proved the theorem, known today as the Rouché-Capelli theorem, which gives conditions for the existence of the solution of a system of linear equations.
- In 1886 Capelli and Garbieri in Corso di analisi algebrica Ⓣ(Course of algebraic analysis) showed that a system of equations having rank k is equivalent to a triangular system with exactly k nonzero diagonal terms.
- In 1886, the year this book was published, Capelli entered the competition for the chair of algebra at the University of Naples.
- In Sulla limitata possibilità di trasformazioni conformi nello spazio Ⓣ(On the limited possibilities of conformal transformations of space) (1886), Capelli gives a geometrical proof of Liouville's theorem on when conformal mappings are Möbius transformations.
- He published Über die Zurückführung der Cayley'schen Operation W auf gewöhnliche Polar-Operationen Ⓣ(On the recycling of Cayley's operation W on common polar operations) in 1887, and then, in Sur les opérations dans la théorie des formes algébraique Ⓣ(On operations in the theory of algebraic forms) (1890), Capelli gives generators for the centre of the enveloping algebra of the Lie algebra GL(n)GL(n)GL(n).
- In papers in 1887 and 1890, Capelli introduced what became known as the 'Capelli identities' which involved equalities of differential operators.
- Finally we mention Capelli's 1901 paper Sulla riduttibilità della funzione xn - A in un campo qualunque di razionalità Ⓣ(On reducing the function xn - A in any rational field).
- As well papers, Capelli published a number of books.
- In the years he spent in Palermo, Capelli taught algebraic analysis for engineers.
- Capelli continued to hold the chair of algebra in Naples until his death.
- In 1894, on the death of Battaglini, Capelli took over the editorship of the Giornale di matematiche which, from that time on, became known as the Giornale di Matematiche di Battaglini.

Born 5 August 1855, Milan, Lombardo-Veneto (now Italy). Died 28 January 1910, Naples, Italy.

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Group Theory, Origin Italy

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive